una mirada a medio siglo de cimentaciones laminares ... · design conditions grant due to the...

A look at half a century of shells foundations, methods ofcalculation and associate research in Cuba

Ángel Emilio Castañeda*1, William Cobelo*, Yoermes González*, José Álvarez*

Una mirada a medio siglo de cimentaciones laminares, métodosde cálculo e investigaciones asociadas en Cuba

* Instituto Superior Politécnico “José Antonio Echeverría”, Ciudad de la Habana. CUBA

Resumen

El trabajo presenta un resumen de las cimentaciones laminares más significativas construidas en Cuba en los últimos decenios y los desarrollos asociados

a estas en cuanto a métodos de cálculo de láminas y cascarones de geometría compleja mediante superficies y cuerpos de referencia, a partir de la

generalización del método de las solicitaciones proyectadas (Pücher, 1934) con el empleo de superficies de referencia (Hernández, 1970) y otros desarrollos

alcanzados en la mecánica del sólido deformable mediante el Método de la Dualidad (Rianitsiyn, 1974; Castañeda, 1993) y la Analogía estático-geométrica

en la mecánica del sólido deformable (Castañeda, 1985). Incluye además una síntesis de las investigaciones desarrolladas en los últimos años sobre el estado

tenso-deformacional del suelo bajo cimentaciones laminares para chimeneas de 74,5 m en Centrales Azucareros (Cobelo, 2004), estudios comparativos

realizados a estas con el empleo del MEF (González, 2010) y otros proyectos investigativos en ejecución actualmente (Álvarez, 2010).

Palabras Clave: Cimentaciones laminares, métodos de cálculo, coordenadas relativas, solicitaciones proyectadas, deformaciones proyectadas

Abstract

The work shows a summary of the most significant shells foundations built in Cuba in the last decades and the developments related to these in term of

methods of calculation of plates and shells of complex geometry using reference surfaces and reference bodies, from the generalization of the projected

solicitations method (Pücher, 1934) with the use of referential surfaces (Hernández, 1970) and other developments made in the mechanics of deformable

solid by the Method of Duality (Rianitsiyn, 1974; Castañeda, 1993) and the Static-Geometric Analogy in the mechanics of the deformable solid (Castañeda,

1985).It also includes a summary of the research developed in recent years on the stress-strain states of soil under and inside shell foundations for chimneys

of 74.5 m in sugar industries (Cobelo, 2004), comparative studies made of these with the use of the FEM (González, 2010) and other research projects

currently running (Álvarez, 2010 ).

Keywords: Shell foundations, methods of calculation, relative coordinates, projected strains, projected deformations

Revista Ingeniería de Construcción Vol. 26 No3, Diciembre de 2011 www.ricuc.cl 245

1 Autor de correspondencia / Corresponding author:E-mail: [email protected]

2 El término “cáscara” o “cascarón” se utiliza en el texto para referirse a estructuras laminares (shells) de espesores medianos y gruesos, dentro de la definición brindada por la IASS(Asociación Internacional de Estructuras Laminares) cuyo comportamiento estructural exige considerar los efectos de fuerzas cortantes y momentos flectores sobre su superficie media,e incluso la posible composición de materiales tipo “sándwich” o de múltiples capas en general que requieran un estudio de distribución de tensiones, solicitaciones o corrimientosen su espesor

1. Introducción 1. Introduction

Fecha de recepción: 14/ 02/ 2011Fecha de aceptación: 01/ 10/ 2011PAG. 245 - 268

Una mirada a medio siglo de cimentaciones laminares/A look at half a century of shells foundations

Structural engineering in Cuba during the secondhalf of the XX century and first decade of the XXIcontains experiences and contributions to the calculationof shells that can not be described without lookingat three very intertwined facts: the design and constructionof shell foundations in buildings and elevatedtanks (need); Relative Coordinates method generalizedthe method of “projected solicitations” on a Cartesianplane for the calculation of plates and shells2 toany surface or body of reference (the solution);

246 Revista Ingeniería de Construcción Vol. 26 No3, Diciembre de 2011 www.ricuc.cl

Ángel Emilio Castañeda, William Cobelo, Yoermes González, José Álvarez

and research on stress-strain behavior of soils andfoundation capacity under revolution shell foundationsin short for chimneys with different numerical models,based on the Finite Element Method (applied research).The paper shows a summary of the most significant resultsof these facts, reveals links and refers to sources for itsstudy and extension.

2. Discussion and development

The work is divided into three essential aspectsof shell structures in Cuba within the period (1955-2010):design and construction of shell foundations buildings(Ruiz, 1962) and elevated tanks (Hernández et al, 1968);the rise of relative coordinates method (Hernandez, 1970)and its subsequent development as a generalization ofthe method of "projected solicitations" (Pucher, 1934),which marked the singularity in the calculation methodsof shell; and research associated with the stress-strainstate of the soil at the hearth and its supportive capacityon frictional soils (C = 0) under revolution shell foundationfor short chimneys (74.5 m). It reveals the identity,difference, links, and dynamics of these aspects, whichrespond achievements, dissatisfactions and new researchtasks that are undertaken.

2. 1 Design and construction of shell foundations inCuba

Shell foundations have met in Cuba over fiftyyears. Shell foundation, as a structural solution forbuildings, chimneys and elevated tanks, enrich theconstructive heritage of the country. However, they lackthe generalization that their economic, construction anddesign conditions grant due to the predominance ofgeotechnical design for stability, given the short durationof the dominant environmental burden (extreme wind)and the lack of significant seismic history.

The use of shell foundation drove the methodsof calculation sheets and shells of complex geometryusing reference surfaces and bodies in a line ofgeneralization of the method of Pucher, and stimulatedresearch on the behavior of layered soils under foundationswith engineering applications.

247Revista Ingeniería de Construcción Vol. 26 No3, Diciembre de 2011 www.ricuc.cl


Thus, in last century fifties it was designed andbuilt in La Havana the first shell foundation as a polygonalshaped raft (folded plate) with braced upper slab tosupport a 24-storey building (94m high) capable ofsupporting winds with a lateral pressure of 300 kg/m2 ona floor with a supportive capacity not exceeding 3 kg/cm2,which has successfully overcome the test of time. Thisfoundation, with a floor area of 2100 m2 (77.2 x 27.2)and transverse bending modulus of 8m and 45cm thick(Figure 1), work of Dr. Ing Sixto Ruiz Alejo (1932),represented a new foundation for the epoch conception,with a 30% decrease in the cost of construction comparedto a solution of raft ribbed trapezoidal beam at fixedprices at the time, not including the benefit of the tanks.(Ruiz, 1962)

Figura 1. Cimentación laminar de placa poligonal (Folded Plate) para edificio alto en Ciudad de la Habana (Cortesía del Dr. Ing.Sixto Ruiz Alejo)

Figure 1. Laminar foundation of folded plate for a high building in Habana City (compliments by Eng. PhD. Sixto Ruiz Alejo)



In addition, the limited availability and high costof timber deposit INTZE type in the country, promoted anew type of elevated tanks for slide mold technology,called type “Güines”, in recognition to the first deposit,designed and build in a shell way, at Catalina de Güines,Habana Province in 1965 (Hernández et al, 1968).

This reservoir of 280m3 capacity and 27m high,was formed by a truncated cone shell foundation ofreinforced concrete with a maximum diameter of 10mmminimum diameter and 2.20 in the shaft, 30cm averagethickness and a basis angle of inclination of 30. Thefoundation was concrete against natural terrain, coupledwith a rotating formwork, and closed at its top and bottomborders by a 70cm thick square ring. It also has a smallcircular plate 60 cm thick in the interior of the shaft. Theshaft is a cylinder of 2.50 m outside diameter and 27min height, constructed by sliding formwork (Figure 2). Thecup and lid were concreted around the shaft before lifting.The new type required a system of metal formwork forconcrete guidelines conical deposits and their coversaround the shaft, until further lifting.

Figura 2. Primer depósito elevado tipo "Güira" con cimiento laminar tronco-cónico, construído eb Catalina de Güines, la Habana,1965 (Hernández y Rubiera, 1968)

Figure 2. First elevated water tank, type “Guira” with laminar trunk-conical foundation, built in Catalina de Guiñez, La Habana,1965 (Hernandez y Rubiera, 1968)

Sección TransversalTransverse section

Superficie inferior delcimiento laminarLower surface of

laminar foundation

Unión fuerte-lámina cónicaJoint shaft-laminar conical

foundation

Detalle del diseñoestructural

Structural design detailSección de la torre

Tower section



The international development of “tower” typestructures from the sixties of XXth century on produceda significant impetus to the shells foundations worldwide,and Cuba was no exception. Hermann Rühle, VicePresident of the International Association of ShellsStructures (IASS) pointed out, at that time: "The foundationbest suited for slender structures radius between 0.1 and0.05 is a closed conical shell with an outer ring. For thissolution chimneys can have the disadvantage of requiringa considerable diameter ... In the case of poor soils andsubsoils foundations circular tower-like structures requirelarge diameter rings. But the shells are the most appropriateway to link the shaft and the outer ring "(Rühle, 1967).

In the late sixties, and in accordance with thisidea, Dr. Ing. José (Pimpo) Hernández (1936-2003)designed and Architect José Licea Delgado (1930-1985)builded in Cuba a revolutionary shells foundation withthe Gaussian shape, following Havelka's proposal fortower type foundations (Leonhardt, 1967), with amaximum diameter of 10.80 m, 2.10 m deep, 20 cmthick and a shaft of the lift tower with 3.00 m in diameter(Figure 3) for a high tank capacity of 200 m3 and 20mheight at the city of Matanzas, which has endured extremewinds of 240 km / h, with gusts to 320 km / h, measuredat the site, structurally unaffected.

The requirements of this foundation, togetherwith the background of the design and construction ofa reservoir as a "water drop" shape for the treatment plant of the city of Santiago de Cuba a few years earlier (Figure3), led Pimpo Hernández to develop, during 1969, anovel method of calculation sheets that generalized themethod of "projects solicitations" on a Cartesian plane(Pücher, 1934) as a method called “Relative Coordinates”,given the possibility of using other reference surfaces andselect the most suitable for the calculation of each sheet,including the possibility that the equation of the platecan be replaced only by a scalar function of the lines ofcurvature of the reference surface in the normal vectordirection (Hernández, 1970). The calculation of shellsfoundations of revolution with guideline of Gaussian formwould not have been possible, at such moment, withouta development of calculation methods such as the onewhich implied this generalization of the method of Pucherusing Relative Coordinates, finally calculated over a polarplane.



The International Conference on "Foundationsfor tower-like structures" developed by IASS in 1970 andthe VIII International Conference on Soil Mechanics andFoundations, 1973, marked a prolific synthesis of researchon this topic. However, experimental studies andnumerical models for problems related to soil-blade, thedistribution of contact pressure between soil andfoundation, the calculation of seat-supporting capacityof soils and the study of zones of contained plasticity bystress concentration effect on building shells on sand,soft clays and soft soils in general were still in its beginingsand identified as a problem of future research.

Whether it was 45cm thick in the "folded plate"of the building in La Habana, the 30 cm of foundationin Guines, or 20cm of the foundation as a bell curve inMatanzas, the shells foundation began its presence instructural engineering in Cuba in unison with itsinternational environment and dragged it the developmentof calculation methods and concerns about the behaviorof the soil, at a time when Nabor Carrillo, former Rectorof the UNAM, prefaced the first books published inMexico on "Soil Mechanics", written by Eulalio Juarez-Badillo and Alfonso Rico.

Figura 3. Cimiento laminar en forma de Campana de Gauss (Matanzas) y depósito elevado en fora de "gota de agua" (Santiagode Cuba) del Ing. José E. (Pimpo) Hernández

Figure 3. Laminar foundation in the shape of a Gaussian Bell (Matanzas) and elevated water tank in the shape of a “water drop”(Santiago de Cuba) by Eng. Jose E. (Pimpo) Hernandez


(1)( 1, 2) = ( 1, 2) + ( 1, 2)

(2)

i Coordenadas Gaussianas/Gaussian Coordinates; Ai Parámetro de Lamé/Lamé Parameter; dSi Diferencial de arco/Differential arc Ri Radio de Curvatura Gaussiana/Gaussian Curvature Radius; i Radio de Curvatura Geodésico/Geodesic Curvature Radius


2.2 The development of the method of relative coordinatesThe method of Relative Coordinates for the

calculation sheet membrane was developed and appliedto cases of roofs, tanks and foundations in a number ofarticles of the “Civil Engineering” Magazine at thebeginning of the decade of the seventies (Hernández etal, 1973, 1974-a, 1974-b, 1975-a, 1975-b) from twobasic ideas (Figure 4-a).

- To define an arbitrary reference surface S of knowngeometry, according to its lines of curvature (1, 2), by the position vector of an arbitrary point P on its

surface as: = ( 1, 2)

- To define the role ( 1, 2) as relative scalarequation of sheet S* to be calculated with respectto arbitrary reference surface S and its lines ofcurvature ( 1, 2), through a distance PP* function, measured in the direction of surface of referencenormal vector S, ( ) until reaching each point S? ofthe real sheet to be calculated.

Whereupon, the equation for the surface S* ofthe real sheet is expressed in terms of lines of curvatureof the reference surface as:

2.1

And all the geometry of the real sheet S* canbe expressed as a function of the Lamé parameters[(A]1,A2) by the unit vectors (e1, e2, en) and its derivativesat the reference surface S, and the relativeequation ( 1, 2) between the two surfaces, withwhich it can be generalize the concept of "designedsolicitations" to relate the arc lengths and unit vectors onboth surfaces by relations of the type:



Where f=0 it means that the reference surfaceS coincides with the average size of the sheet S* to becalculated in lines of curvature, which causes that forthat Ki= 1, Bi = i=0 matter would 1= x; 2 = y; y f ( 1,

2) = z (x,y) coincide with Pucher, and the referencesurface would be a Cartesian plane (x,y).

The method of Relative Coordinates opened atleast three new ways to calculate shells structures:

- It established the geometric base for generalizingPücher's ideas in relation to the introduction of“projected solicitations” on a Cartesian plane, creatingthe possibility of selecting the surface of referenceS more suitable for the calculation of each shellaccording to its shape, charge and/or conditions ofsupport (Example: polar plane or a cylinder coaxialfor a shell of revolution with loads in the directionof the vector o ; Cartesian plane for thecalculation of a rectangular base hyperbolicparaboloid, parallel to its asymptotes, etc.)

Figura 4. Coordenadas Relativas entre dos superficies (4 a) y entre dos medios tridimensionales (4 b)Figure 4. Relative coordinates between two surfaces (4a) and between two tridimensional means (4b)



- It provided a new way to calculate shells of complexshape without analytical expression to describe theiraverage size, setting, through analytical or numericalway, an equation concerning the distances betweenthe medial surface of the shell S* and an area ofreference known as S by a function type ( 1,

2)- It extended the possibilities of calculating in "projected

solicitations", boundary problems having a simplergeometric shape on a reference surface S than onthe actual average surface S? of the shell, as it is thecase of two cylinders of equal diameter, that cut theirlong axes at 90 degrees, and whose curve ofintersection in warped space, projected as a contour,a straight diagonal line over a Cartesian plane,selecting S as a reference surface for its calculation.

A test of the potential of this geometric approacharose almost simultaneously in the second half of theseventies when a group of researchers from the KazanAviation Institute (Paimuchin et al., 1975, 1977, 1980),without using Pücher's method of "projected solicitations",proposed a similar geometric equation for the approximatecalculation of shells lowered and set the basic equationsfor calculating shells of complex shapes such as "flashlights"and other aircraft parts from this geometric perspective,with linear and nonlinear approaches for the calculationof shells of "sandwich" type of constant and variablethickness by calculating approximate surface.

While in Cuba, since the mid-seventies untilthe nineties, continued research on the method of relativecoordinates with ¨ projected solicitations ¨ and new resultswere achieved that allowed the planning of the relativecoordinates with traditional methods of differentialgeometry of shells not referred to lines of curvature(Castañeda, 1984-a) and its generalization for thick andmedia shells, which can not be reduced to an averagesize (Castañeda, 1985, 1995), through the introductionof the equation between two three-dimensional mediaS z and S z*, and developments resulting in differentialgeometry (Figure 4-b),



Where the geometry of the real three-dimensional media, Sz* , is based on thegeometry of the referential three-dimensional Sz*

media for everything: z = z* D; o z = z*/Dproviding that:

In these works (Castañeda, 1985, 1995) wereestablished the conditions of orthogonality of the unitvectors in the average surface of the shell by its equationof relative coordinates ( 1 = 0, o 2 = 0 , associated to( = 90°); the case where the conditions for lowering ashell on its surface when not required solicitations andprojected deformations (si 1 y 2 if and tend to zerosimultaneously), V.N. Paimuchin correcting criteria onthe conditions of orthogonality and of debasement ofthese shells in relation to their surface calculation andestablish geometric relations in relative coordinates foran orthogonal reference system, in correspondence withthe geometric criteria for a point of a thick shell(Goldenveizer, 1953).

Thus, membrane theory included the ¨ projecteddisplacements "(Castañeda,1981-a) and "projecteddeformations ¨ (Castañeda,1982-a, 1982-b, 1982-c) tomove the whole process of calculation to the referencesurface on hiperstatic problems, supported by the methodof dual static-geometric equations (Rianitsyn, 1974) andthe method of virtual work (Hernández, 1982), creatingconditions for obtaining the basic equations of the theoryof flexion of Love-Kirchhoff type and Timoshenko typein Relative Coordinates to solicitations, deformationsand displacements projected on the referencesurface (Castañeda, 1983-a, 1983-by 1983-c),

(3)

(4)



which was vital to provide integration andcomplementation to the method where "no membrane"extend the results to structural laboratory experiments inphysics (1984-b), and form a static-geometric view pointthat generalizes Pücher's method for any reference surfacemembrane theory, bending and analysis of thick shells,which was the most significant result of this stage in thedevelopment of methods of calculation (Figure 5).

The equilibrium equations (2.5), geometric (2.6),physical or constitutive (2.7), of projected solicitationsto real solicitations (2.8) and deformation projected toactual deformation (2.9) in Relative Coordinates to thetheory of bending type Love-Kirchoff in which the straightand normal element to the shell middle surface remainsstraight and normal after deformation , has for (f = 0) theclassic cases of Gauss intrinsic coordinates, and for 1=

x; 2 = y; y f ( 1, 2)= z(x, y) Pücher's equation showingthe validity of this approach and the new possibilitiescreated to establish and solve boundary problems (polarplane, cylindrical, coaxial, etc..) only by introducing thegeometric parameters of the first and second quadraticform if it has a function of the type ( 1, 2).

Furthermore, the method of Duality (Table 1),applied between the equilibrium equations (2.5) and thegeometrical equations (2.6), confirmed the relationshipbetween the projected displacement vector (linear andangular) as a result of neglecting deformations shear (Qi)in Love-Kirchhoff Theory and consider the infinite stiffnessof the shell with respect to the torsion of the normal axisto its average surface , as it corresponds to thisapproach.



Figura 5. Modelación del Método de Pücher en Coordenadas Relativas para los diferentes niveles de análisis de cáscaras: a)Membrana; b) Flexión; c) Cáscaras gruesas.

Figure 5. Modeling of Pucher Method in Relative Coordinates for different levels of shells analysis a) membrane; b) flexion;c) thick shells



Equilibrium equations in projected solicitations

2.5

Where:

q1; q2; qn; m1; m2: Forces and moments are mass flow

in the direction Q1 y Q2: Are the projected shear

Geometric equations projected deformations anddisplacements

(5)

(6)


E: Módulo de elasticidad/Elasticity modulus; : Coeficiente de Poisson/Poisson ratio; h: Espesor/ThicknessNi, Nj: Solicitación normal proyectada/Projected normal solicitation; Nij: Solicitación tangencial proyectada/Projected tangential solicitationMi, Mj: Solicitación de torsión proyectada/Projected torsional solicitation; Mij: Solicitación de flexión proyectada/Projected bending solicitation

i* j: Deformación longitudinal proyectada/Projected longitudinal deformation; ij: Distorsión proyectada/Distortion projected

i, j: Deformación de torsión proyectada/Projected torsional deformation; ij: Deformación de flexión proyectada/bending deformation projected


Constitutive physical equations and deformationsprojected solicitations.

Relationship among projected solicitations and actualsolicitations

Relationship among projected deformations and actualdeformations

The method of Relative Coordinates seeks newdevelopments today pointing to the obtention ofCompatibility Equations of Deformity type Saint Vennatand Airy stress function, Maxwell or Morera throughDuality method (Rianitsyn, 1974) and static -geometricanalogy (Castañeda, 1985, 1993) for the solicitations anddeformations calculation projected by using semi-inverseand inverse methods associated with the hiperelesticitydegree of the shells, with Finite Difference solutions,where the operator transpose of the ordinary differentialequation:

(8)

(9)

(7)



and the transposed operator of differentialequation in partial derivatives:

When a (0), a (1)… a (m), and b(0), a (1)… b(m), are functions of the different variables and (k N), whichallows to obtain the geometric equations transposingequations of equilibrium (Rianitsyn, 1974) as shown atTable 1 introducing equilibrium equations per rows (2.5)and extracting per column geometric equations (2.6).

2.3 Research of stress-strain state of soils underrevolutionary shells foundation for Finite ElementMethod (MEF)

In the first decade of XXI century Cubahas done research on models based on FiniteE l e m e n t M e t h o d ( S I G M A / W , P L A X I S ,

Tabla 1. Relación transpuesta del método de la dualidad para la teoría de Love-KirchoffTable 1. Relationship transpose duality method for the Love-Kirchoff theory

tiene la forma

(10)

(11)



ABAQUS) on the behavior revolution shells foundations(straight and parabolic guideline) (Figure 6) and foundationsoils under laminar axial-symmetric load (Cobelo, 2004,Gonzalez, 2010) for the determination of the contactpressure distribution, seating, supportive capacity of soilsand solicitations in the structure, considering the soil-blade interaction. This work was carried out in threestages of research as an alternative foundation for shortstacks of 74.5 m in height, typical Cuban structure forsugar mills.

The studies for these fireplaces were done ona foundation geotechnically designed of 16m diametercomposed of two layers: an inverted dome 8m at thebottom of shaft, which works entirely in compression,and a truncated cone plate embedded in the shaft andfree at the outer edge of the foundation. The truncatedcone sheet in all cases was divided into 10 cross sections3, 5 and 9 points defined on each section with whichwere evaluated the handle radial (Nr) and circumferential(N ) in the middle surface. When structural concrete wasgiven a behavioral model linear-elastic, withE=2,16x107kPa and Poisson Coeficient =0,176.

In the first stage (Cobelo, 2004) it was developedan experiment 32 with permutations of soils in the core,below the floor slab; value of the angle of internal friction

' ( '1=20º, '2=30º y '3=40º) and Elasticity Moduls(E01=7260kPa, E02=11000kPa y E03=30000kPa), withthree conical geometries, constructively competitive anddifferentiated by the angle ( ) of the guideline inclinationin reference to the base plane (26,5 º, 35 º y 45 º) and a relationf/a (0,5; 0,7 y 1,0), determining 27 numeric experiments,

Figura 6. Características generales de la modelación de la cimentación por el MEFFigure 6. General characteristics of foundation modeling by MEF



done with GeoSlope v5.13 SIGMA/W Software, repetedfor elastic and elasto-plastic models (Mohr-CoulombType) of soil behavior, up to a total of 54 studied cases.

In the second stage (González, 2010) anothernumerical experiment was raised where 45 cases wereconsidered of homogeneous soils in the core, below thefloor slab for the same values of friction angle ' andModule elasticity of soils E0i for the three previous conicaltrunk geometries and two new parabolic geometries ofnegative Gaussian curvature (the worst) with the use ofPlaxis 2D v8.2, to confront them with the results of thefirst stage.

In the third stage, still running, new numericalexperiments are performed for all permutations of soilsin the core, below the floor slab level with models ofelastic and elasto-plastic behavior, using ABAQUS programon sheets of revolution of double curvature of positiveGaussian curvature, increasing the number of nodes oneach cross section of the sheet, which are compared withthe results of inverse solutions in Relative Coordinatesand Theory of Flexion Love-Kirchhoff type by theapplication of Finite Differences method.

In reference to radial solicitations (Nr) andcircumferential (N ) in the average surface of the sheet(Figure 7) the results of the first two steps show that theinfluence of a type or another of frictional soil and itsdisposition in one or another of the identified zones hasless significance, for the same geometry, than the elasticor elasto-plastic model of physical constitutive behaviorasumed, and that such influence is even minor in themeasure that the sheet augments its relation f/a. The firsttwo graphs Nr y N appearing at the top of Figure 7for f/a = 0,5 (26,5º) correspond to a linear-elastic behaviorof the soil, the next two ones to an elasto-plastic behaviorof Mohr-Coulomb type for the same geometry, and thetwo last ones to the same lineal-elastic model for f/a =1(45º). In each graph are shown the variations of Nr yN for the studied 9 combinations of soils, confirmingits characteristic behavior in these cases.



Figura 7. Gráficos de Nr y N para diferentes relaciones de f/a y modelos de suelos

Figure 7. Graphs for Nr and N for different f/a relations and soil models



The results obtained by computer modelingwith application of MEF considering soil-structureinteraction, allowed to enter "correction factors to themembrane solicitations" Fc

r and Fc and the Effect ofSimple Alteration.

The vertical stress distributions ( y) at the sill,corresponded in all cases to Szechy physical tests (1965),Nicholls and Izadi (1968) and Kurian (1983), confirmingthat there is vertical stress concentration towards thelayered conical edges of the foundation and a plasticredistribution beneficial for the central area (Figure 8).

Vertical displacements (sy) on the foundationsill showed that the soil confined core is not incompressibleand absorbs vertical deformation in part of the sill area,this being more pronounced toward the outer edge, wherethere are areas of contained plasticity near the end of therolled edge ring (Figure 9).

Figura 8. Distribución de tensiones en la solera para suelo elástico y elasto-plástico en f/a = 0,5Figure 8. Curb stress distributions for elastic soil and elastic-plastic soil in f/a = 0.5

Figura 9. Distribución de asientos (sy)en la solera para suelo elástico y elasto-plástico en f/a = 0,5Figure 9. Curb foundation distribution (sy) for elastic soil and elastic-plastic soil in f/a = 0.5



Studies of qu, load capacity of the soil underthe shell foundation analyzed show that these values canbe achieved up to 40% higher than load capacityTerzaghi's equation, 34% higher than those determinedby Hansen's criteria, and up to 20% calculated onMeyerhof's criteria, depending on the angle of inclinationof the guideline and soil types present in the nucleus, thefiller and the base. Table 2 shows the qu load capacityresults obtained by FEM for geometric variation f/a=0,7in comparison with Terzaghi, Hansen and Meyerhofanalitic methods, depending on the soil friction angle.

Thus, and similar to other researchers (Rahman,1987) was estimated a correction factor Fq that modifiesthe coefficient Nq calculated by Reissner (1924) [Bracha(2000), Cuban Standard for Geotechnical Design ofSuperficial Foundations (2007)] reconciles the values ofthe capacity of frustoconical foundations with thosedetermined for displaced shell foundation of triangularsection (Hanna y Rahman, 1990), with a new correctionfor the effect of soil-shell. In addition, Ss established formfactors that added to the equation of Terzaghi load capacity(with load capacity coefficients corrected by the factorFq), taking into account the influence of the shape of thefoundation in the increase of qu (Table 3).

Tabla 2. Resultados de capacidad de carga qu en modelo de MEF y métodos analíticos en f/a = 0,7Table 2. Results of loading capacity qu in MEF model and analytical methods in f/a = 0.7

Tabla 3. Coeficientes de forma corregido Ss para la capacidad de carga bajo láminas tronco-cónicaTable 3. Amended shape coefficient Ss for loading capacity under trunk-conical plates



In the second stage of the investigations it wasused an "interface" with an elasto-plastic Mohr-Coulombtype behavoir, affected by a Rinter = 0.666 coefficient[Ibañez (2000) and Cobelo (2004)], over all the contactsurface between soil and foundation plate. Based on theexperiences of other researchers [Huat & Mohamed (2006)and Esmaili & Hataf (2008)] and the use of Plaxis 2D(Gonzalez, 2010) were modified and expanded theprevious models, taking advantage of this tool: automaticgenerator and unstructured meshes and particular dataoutput of the load-deformation relation in a virtual trial-load capacity through the calculation option "c- ''reduction" which established other graph criteria for thedetermination of ultimate load capacity. The results ofcalculation according to this criterion are presentedgraphically (Figure 10) for each depth of foundation Df,and qu curves are included according to Terzaghi andHansen criteria [Braja (2000), Cuban Standard forGeotechnical Design of Superficial Foundations (2007)]for foundations of flat circular equal base area, as theone obtained in the first stage for the truncated conesheet of straight guideline f/a = 07.

The research planned in the third stage, withthe ABAQUS program and the use of semi-inverse methodsin Relative Coordinates, Theory of Finite Differences andFlexion, plans to introduce the effect of the HorizontalForce and the eccentricity of the vertical load oncomputational modeling shell foundations of these shortstacks.

Figura 10. Resultados de capacidad de carga qu para diferentes Df , según método “c- ' reduction”Figure 10. Results of loading capacity qu for different Df , according to “c-´ reduction” method



3. Conclutions

The past half century shell foundations enrichedthe built heritage in Cuba and the engineering professionalculture of the country, developing new methods ofcalculation and research partners, as part of an overallglobal reach process. Among the conclusions of thisperiod there are three main aspects:

- The good structural behavior of shell foundations ofelevated tanks built in Cuba in the early 60's of lastcentury coincides with the beneficial results fromthe point of view of pressure distribution, settlementsin the hearth and ability load level as more recentresearch show in tower-like structures (short chimneys)with geotechnical and computational models basedon the application of the Finite Elements Method(FEM). The distributions of stresses and hearthsettlements obtained by this method show that thesoil contained within the shell is considerablydeformable, as posed by the design assumptions ofthat time, because it absorbs some of these tensionsand redistributes better stresses and strains withrespect to a circular rigid base foundation.

- In the range of geometries analyzed (guidelines to=26,5º (f/a=0,5) and =45º (f/a=1,0)), constructively

acceptable for this type of shells, the results on thedeviations in the physical properties of soil constitutivefrictional (c = 0), confirm that these variations do notintroduce significant differences, for engineeringpurposes, on the value of the solicitations Nr y Ndepending on the angle of internal friction of soil.Furthermore, these results show that small variationsin the shell guideline in the construction technologyused (in situ or precast) do not introduce largemodifications in the internal solicitations of it, whilethe degree of compaction of soils within the shelland over the hearth level, which are variables thatcontrol the engineer during the execution of thework, may have greater significance in the stressdistribution and settlement of the foundation level.

4. Referencias / References



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Castañeda Hevia A. E. (1982-b), Teoría membranal de las cáscaras en coordenadas relativas con deformaciones proyectadas.¨Ingeniería Civil, Vol. XXXIII, No 5. La Habana, Cuba.

Castañeda Hevia A. E. (1983-a), Teoría general de las cáscaras elásticas en coordenadas relativas con deformaciones proyectadas.(1ra parte), Ingeniería Civil, Vol. XXXIV, Nº 1, enero-febrero, La Habana. Cuba.

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